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Minority Games
A Complex Systems Project
Going to a concert…
• But which night to pick? Friday or Saturday?• You want to go on the night with the least
people. So you try and guess which night.• You can base your guess on previous
attendence. But so does everyone else.• This is a minority game.• What happens if the concert is on every
night?
Properties of Minority Games
• Participants try to pick the least common choice.
• Communication between participants is only through results of previous attempts.
• Each participant thus makes decisions based upon their private strategies and the public history.
• Example: The Stock Market
Why are Minority Games Complex Systems?
• Large numbers of agents, each with their own strategy sets.
• The system adapts to new information each round.
• The history is important.
• The system is frustrated: the more successful a strategy is, the worse it gets.
Choose a Number!
• Pick an integer number between 1 and 10, and try to get the least common number.
• We used the first year physics class as our system.
• The experiment was repeated, but each time the results of the last round were left up.
• 6 rounds were run in total.
Results (First Four Rounds)
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10
Round 1
0
5
10
15
20
1 2 3 4 5 6 7 8 9 10
Round 2
0
5
10
15
1 2 3 4 5 6 7 8 9 10
Round 3
0
5
10
15
20
1 2 3 4 5 6 7 8 9 10
Round4
The jaggedness Parameter (a)a histogram
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10number
freq
uen
cy
a=57.4
distribution of variance (a), for Random number choice
0
0.05
0.1
0.15
0.2
0.25
0.2
2.17
4.13 6.1
8.07
10.2
12.2
14.1
16.1
18.1
20.2
22.1
24.1
26.1
28.2
30.2
32.1
34.1
36.1
38.2
41.1
a
P(a
)
1 23 4
Our Models
• 100 agents
• Each agent chooses a number according to a strategy
• Calculate histogram of results
• Rank each number from 1-10 based on popularity.
number 1 2 3 4 5 6 7 8 9 10frequency 8 9 13 6 5 21 6 11 9 12rank 4 6 9 2 1 10 3 7 5 8
Example
• Strategies got 1 point if they led to the least popular number and 0 otherwise.
Model 1distribution of varience
0
200
400
600
800
1000
1200
14000 7 14 21 28 35 42 49 56 63 70 77 84 91 98
varience
freq
uen
cy
Distribution of Variance
Variance
varience with time
0
20
40
60
80
100
120
0 20 40 60 80 100time (round number)
varie
nce
Variance with Time
Model 1distribution of variance (a), for Random number choice
0
0.05
0.1
0.15
0.2
0.25
0.2 2.17
4.13 6.1 8.07
10.2
12.2
14.1
16.1
18.1
20.2
22.1
24.1
26.1
28.2
30.2
32.1
34.1
36.1
38.2
41.1a
P(a)
distribution of varience
0
200
400
600
800
1000
1200
1400
0 7 14
21
28
35
42
49
56
63
70
77
84
91
98
varience
freq
uen
cy
Variance
Distribution of VarianceV
aria
nce
What is happening here?
0
2
4
6
8
10
12
14
1 2 3 4 5 6 7 8 9 10
strategy Mostpopular
2nd 3rd 4th 5th 6th 7th 8th 9th Leastpopular
score 0 0 1 0 0 0 0 0 0 0
a=5.6
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10
Previous round
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10
What is happening here?
0
2
4
6
8
10
12
14
1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
1 2 3 4 5 6 7 8 9 10
strategy Mostpopular
2nd 3rd 4th 5th 6th 7th 8th 9th Leastpopular
score 0 0 1 0 0 0 0 0 0 1
Model 2Before the Trial:
• Agents given patience parameter, p.
During the Trial:
•Agents choose a number and stick with it.
•Every p rounds, consider changing.
•When changing, change to best number of previous round.
And then this happened…
Insert alpha distribution.
Distribution of a - model 2
01000
20003000
40005000
6000
0 4 8 12 16 20 24 28 32 36 40 44 48 52
a
Fre
qu
en
cy
distribution of variance (a), for Random number choice
0
0.05
0.1
0.15
0.2
0.250.
2
2.17
4.13 6.
1
8.07
10.2
12.2
14.1
16.1
18.1
20.2
22.1
24.1
26.1
28.2
30.2
32.1
34.1
36.1
38.2
41.1
a
P(a
)
What should have happened…
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10
What should have happened…
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10
What should have happened…
Time series for a - model 2
0
2
4
6
8
10
12
1 3 5 7 9
11
13
15
17
19
21
23
Round
a
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10
What went wrong…
02468
101214161820
1 2 3 4 5 6 7 8 9 10
What went wrong…
What went wrong…
Time Series for a - model 2
0
10
20
30
40
27
77
0
27
78
2
27
79
4
27
80
6
27
81
8
27
83
0
27
84
2
27
85
4
27
86
6
27
87
8
27
89
0
27
90
2
27
91
4
27
92
6
27
93
8
27
95
0
27
96
2
time
a
Model 3During the Trial:
•Agents choose a number and stick with it.
•Every round, agents have a probability .02 that they will consider changing.
•When changing, change to best number of previous round.
This time it worked!Distribution of a - Model 3
0
10000
20000
30000
40000
50000
1 3 5 7 9
11
13
15
17
19
21
23
a
Fre
qu
en
cy
distribution of variance (a), for Random number choice
0
0.05
0.1
0.15
0.2
0.250.2
2.1
7
4.1
3
6.1
8.0
7
10.2
12.2
14.1
16.1
18.1
20.2
22.1
24.1
26.1
28.2
30.2
32.1
34.1
36.1
38.2
41.1
a
P(a
)
Good for wealth distribution too.Wealth distribution
0
5
10
15
20
4000
4066
4150
4250
4350
4450
4550
4650
4750
4850
4950
5050
5150
wealth after 50000 rounds
nu
mb
er
of
pe
op
le
Average wealth = 4542
wealth distribution under random choice
0
2
4
6
8
10
2590
2610
2630
2650
2670
2690
2710
2730
2750
2770
2790
2810
2830
2850
2870
2890
wealth after 50000 rounds
nu
mb
er o
f p
eop
le
Average wealth =2733
Minority Game Theory
Formalism
2 choice minority game- extension of the El Farol Bar problem
N players, two choices: 0 or 1
Rules: At each turn, every agent chooses a side (0 or 1), those that end up in the minority side, win.
Formalism
• Memory: the bit-string of past winning outcomes.
eg. minority side trial one – 1, trial two -1, trial three – 0, trail four - 1, trial five - 0.
M = {1,1,0,1,0} with length: m = 5.
Strategy Space
History Prediction
000 0
001 0
010 0
100 0
011 1
101 1
110 0
111 1
History Prediction
000 0
001 1
010 1
100 0
011 0
101 1
110 1
111 1
Strategies defined in an abstract way - no psychology.
Strategy s – a ‘card’ with a prediction for each possible past history.
History Prediction
000 1
001 0
010 0
100 1
011 1
101 0
110 0
111 0
eg. Strategy1 Strategy2 Strategy 3 etc…
Minority Game Structure
• Odd number (N) of agents, each given two or more strategies selected at random from the strategy space.
• At each turn, the strategies are evaluated; a point is awarded to each strategy that predicts the correct minority result. The strategy with the highest number of points is chosen for the next turn.
• Evaluated for large number of time steps, for different memory lengths m.
Complex behaviourSavit et al, 1999.
Region 1:
Small number of strategies, crowded behaviour – similar to model 1 of the MCMG.
Region 2:
More available strategies. Able to cooperate to achieve low variance.
Region 3:
Very large strategy space. Less probability of cooperation.
Conclusions• Interesting behaviour
• Possible to do better-than-random
but with abstract/slightly contrived strategies
• Further study to see the stability of better-than-random systems with plausible strategy sets.
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